A harmonic oscillator system with the generalized commutation relations

Abstract

The system of a one-dimensional harmonic oscillator is discussed with the generalized commutation relations obtained by Wigner. The Hamiltonian is shown to have self-adjoint extensions. The domain of self-adjointness is explicitly specified in some cases. The proof is carried out by the use of the Lax–Milgram lemma. A suitable rigged Hilbert space is found for this system to reformulate the earlier arguments. The main difficulty is that the momentum and Hamiltonian operators contain the reflection Rψ(x)=ψ(−x) as well as singular terms.